An efficient method for optimizing nested open pits with operational bottom space

Determining a set of nested pits to support the design of an open pit mine that leads to high economic value is crucial for the strategic planning of these operations; thus, practitioners rely on optimization methods for finding high-value solutions. However, current approaches are not sufficient as they lack at least one of the following features: fast computations of optimal solutions, good geometric properties, and nestedness of the pits. In this work, we propose an optimization model to address the problem of determining multiple nested pits by introducing a cost-based penalty for not meeting precedence constraints linked to a minimum bottom width. Using penalties instead of constraints is novel and turns out to have several advantages. First, the constraint matrix is totally unimodular; thus, the problem can be solved efficiently. Second, the model can be parameterized to generate nested pits. Therefore, our model is the first published model that is efficient, can be solved to optimality, preserves the nestedness of the solutions, and produces geometries more amenable for mine design, without the need for heuristics. Finally, we devise an iterative method that profits from the nestedness of the solutions to speed up the resolution and test the model in three different data sets, with different geometrical and cost parameters for a total of 135 different instances. The results show that the geometry of the bottom pits is indeed improved and that we can solve the problems up to optimality up to 80% faster than an off-the-shelf solver.